Lifetime Moments

For the expected survivor time of the joint life status, we have
begin{eqnarray*}
dot{e}_{xy} = E ~ T(xy) &=& int_0^{infty} t f_T(xy)(t) dt =
int_0^{infty} (1-
F_T(xy)(t)) dt \
&=& int_0^{infty} ~_t p_{xy} dt .
end{eqnarray*} The third equality comes from an integration by parts.

Similarly, for the expected last-survivor time,
begin{eqnarray*}
dot{e}_{overline{xy}} &=& E ~ T(overline{xy}) = int_0^{infty}
~_t p_{overline{xy}} ~ dt \
&=& int_0^{infty}
left(~_t p_x + ~_t p_y – ~_t p_{xy}right) dt \
&=& dot{e}_{x}+dot{e}_{y}-dot{e}_{xy}.
end{eqnarray*} Note that this relation holds without regard to the independence between lives assumption.

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